4-D Inversion Of Geophysical Data And 4-D Imaging Method Of Geological Structure Using It

ABSTRACT

The method for 4-D inversion of geophysical data for calculating distribution of subsurface material properties from geophysical data includes (a) defining measured data into space-time coordinates, and defining a reference space-time model vector (U) composed of many reference space model vectors (U) for a plurality of pre-selected reference times to simulate a space-time model vector (P) that is a geologic structure continuously changing in time; (b) approximating a numerical modeling for a geologic structure space model at an arbitrary time using Taylor series of numerical modeling for the reference space models, defining an objective inversion function to constrain each inversion in space and time domains, and obtaining a reference space-time model vector (U) from the measured data defined in space-time coordinates using the objective inversion function; and (c) obtaining a space-time model vector (P) from the reference space-time model vector (U) to calculate distribution of subsurface material properties changing in time.

TECHNICAL FIELD

The present invention relates to 4-D inversion of geophysical data and a4-D imaging method of geologic structure using it, and more particularlyto a new 4-D inversion and an imaging method using it, which obtains thedistribution of subsurface material properties in the space-time domainusing just one inversion for monitoring data at different times so as toobtain information related to geologic structure changing in time.

Geophysical monitoring advantageously allows obtaining 2-D or 3-D spaceimage of an examination area, not information of one point. Due to thisreason, the geophysical monitoring conducting many investigations in thesame region with changing measurement times is more frequently used tounderstand the change of geologic structure changing in time. Thegeophysical monitoring is applicable to very wide fields such asmonitoring the change of foundation of an important structure,monitoring dispersion of pollution, monitoring the subsurface disaster,determining the foundation improvement, and deriving hydrogeologicalmaterial properties such as passage of underground water, hydraulicconductivity, permeability and porosity.

BACKGROUND ART

In a common analysis of monitoring data, survey data obtained atdifferent times are independently inverted to obtain images of eachtime, and then the images are compared with each other to interpret andmonitor the change of foundation along with time, However, in mostcases, this approach has high probability of erroneous analysis due toinversion artifacts since the subsurface material properties are notgreatly changed.

FIG. 1 is a concept view showing a conventional inversion for imaginggeologic structure changing in time.

In the conventional inversion, monitoring data were separately handledto realize subsurface images, so it is impossible to refer to geologicstructure changing at different times. Thus, the subsurface images maybe seriously distorted, and the distortion appearing in the images ofchanging geologic structure gets worse. In addition, since eachmonitoring data and subsurface space are analyzed just in a spaceconcept, it is impossible to consider the change of geologic structurethat may happen while measuring the monitoring data.

In order to prevent the above problems, an inversion method forinverting monitoring data by adopting an inversion result of initialdata as a prior model or reference model (Locke, 1999; Labrecque andYang, 2002) has been suggested. However, this method is not differentfrom the conventional inversion in the point that conversion formonitoring data are independently conducted, and also it is apparentthat the entire image while the monitoring period is very highlydependent on the initial model.

Meanwhile, the most basic assumption commonly used in the inversion forimaging geologic structure is that the geologic structure is not changedwhile the data is monitored. However, in case of brine injectionexperiments on a soil layer having very high permeability or the like,fluid with very high electric conductivity migrates fast, so the aboveassumption for the static subsurface model can be hardly accepted. Inthis case, if the factor that the geologic structure changes whileobtaining data is not included in the inversion, the subsurface imageobtained by the inversion will be severely distorted as apparentlyunderstood. In spite of that, most of studies and techniques developedso far are based on the assumption that geologic structure is notchanged.

Meanwhile, Day-Lewis et al. (2002) has developed an effective inversionfor crosshole radar tomography monitoring data obtained from thegeologic structure changing in time while obtaining the data. However,this inversion is applicable only for travel time tomography that uses atravel time of wave, and it is substantially impossible to expand it toinversion of physical survey data.

DISCLOSURE Technical Problem

In order to solve the above problems, the present invention is directedto providing a new 4-D inversion capable of: i) obtaining a subsurfaceimage of far higher reliability than conventional ones by inverting aplurality of monitoring data; and ii) calculating a subsurface imagecontinuously changing in time using only one measured data as well as aplurality of iterative measured data.

Technical Solution

In order to accomplish the above object, the present invention providesthe method for 4-D inversion of geophysical data for calculatingdistribution of subsurface material properties from geophysical data,which includes (a) defining measured data in space-time coordinates, anddefining a reference space-time model vector (U) composed of a pluralityof reference space model vectors (U) for a plurality of pre-selectedreference times so as to simulate a space-time model vector (P) that isa geologic structure continuously changing in time; (b) approximating anumerical modeling for a geologic structure space model at an arbitrarytime using Taylor series of a numerical modeling for the plurality ofreference space models, defining an objective inversion function toconstrain each inversion in space and time domains, and obtaining areference space-time model vector (U) from the measured data defined inspace-time coordinates using the objective inversion function; and (c)obtaining a space-time model vector (P) from the reference space-timemodel vector (U) to calculate distribution of subsurface materialproperties changing in time.

Using the 4-D inversion of the present invention, it is possible torealize reliable subsurface images even when the change of geologicstructure happening during data collection cannot be ignored.

While measurement data were defined only in space coordinates in theconventional inversion, the measurement data are defined in space andtime coordinates, namely 4-D coordinates, in the 4-D inversion of thepresent invention.

The method for 4-D inversion of the present invention is conducted basedon the assumption that subsurface material properties are linearlychanged in time. Thus, a space vector p(t) at an arbitrary time t isdefined using reference space model vector u at a pre-selected referencetime as in the following equation, and thus the space-time geologicstructure model continuously changing in time may be simulated intoseveral reference space models.

p(t)=u _(k)+(t−τ _(k))u _(k)  [Equation]

Here,

${v_{k} = \frac{u}{t}},$

and τ is a pre-selected reference time.

In addition, in the step (b), numerical modeling G(t) for the geologicstructure space model at an arbitrary time τ_(k)≦t≦τ_(k+1) is conductedin a way of approximating by using the first order Taylor seriesexpansion of the following equation for the numerical modeling ofreference space models at two reference times (τ_(k), τ_(k+1)).

$\begin{matrix}{{G(t)} = {{\frac{\tau_{k + 1} - t}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k} \right)} + {\left( {t - \tau_{k}} \right)J_{k}\upsilon_{k}}} \right\}} + {\frac{t - \tau_{k}}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k + 1} \right)} + {\left( {t - \tau_{k + 1}} \right)J_{k + 1}\upsilon_{k}}} \right\}}}} & \lbrack{Equation}\rbrack\end{matrix}$

Here, F(u_(k)) is numerical modeling for the reference space modelu_(k), and J_(k) is partial derivatives of a model response with respectto the reference space model.

Using the steps (a) and (b), the subsurface model in space and timedomains continuously changing in time is approximated into a pluralityof reference space models, and the problem of numerical modeling fornumerous space models requiring a lot of calculation is also solved in away of approximation conducted by the modeling based on reference spacemodels.

Inversion is defined using an objective function to be minimized duringthe iterative inversion process, and the 4-D inversion of the presentinvention is also defined using the objective function of the step (b).The objective function of the 4-D inversion of the present invention iscomposed of three terms; an error between measured data and simulateddata for a space-time model, a constraint regarding a space domain, anda constraint regarding a time domain. The constraint regarding spacedomain employs a smoothing constraint that geologic structure issmoothly changed in aspect of space. One of essential features of the4-D inversion of the present invention is to introduce a constraint fortime domain to the inversion, and for the newly introduced time domain,it is preferred to introduce a constraint that there is no great changebetween two reference space models adjacent to each other in aspect oftime. This objective function of 4-D inversion is defined using thefollowing equation.

Φ=∥e′∥ ²+λΨ+αΓ  [Equation]

where,

${e^{\prime} = {d - {G\left( {U + {\Delta \; U}} \right)}}},{\Psi = {\left( {{\partial^{n}\Delta}\; U} \right)^{T}\left( {{\partial^{n}\Delta}\; U} \right)}},\begin{matrix}{\Gamma = {\sum\limits_{i = 1}^{m - 1}{{\left( {u_{k} + {\Delta \; u_{k}}} \right) - \left( {u_{k + 1} + {\Delta \; u_{k + 1}}} \right)}}^{2}}} \\{{= {\left\{ {M\left( {U + {\Delta \; U}} \right)} \right\}^{T}{M\left( {U + {\Delta \; U}} \right)}}},}\end{matrix}$

Ψ is an inversion constraint function in a space domain, Γ is aninversion constraint function in a time domain, λ and α are Lagrangianmultipliers for controlling the degree of constraints, d is a measureddata vector, G is a numerical modeling result or a numerical simulationresult for a given geologic structure, U is a reference space-time modelvector composed of a plurality of reference space model vectors (u), andM is a square matrix where diagonal and one sub-diagonal elements havevalue 1 or −1.

In addition, in order to accomplish the above object, there is alsoprovided a method for 4-D imaging, which further includes the step ofimaging geologic structure changing in time, based on the distributionof subsurface material properties obtained by the above 4-D inversion.

ADVANTAGEOUS EFFECTS

According to the present invention described above, it is possible toprovide a new 4-D inversion and an imaging method using it, which allowsto accurately calculate geologic structure changing in time by invertinga plurality of monitoring data at the same time, and provide reliableimages even when the geologic structure changes fast during datacollection, using even only one measurement data, not a plurality ofiterative measurement data.

DESCRIPTION OF DRAWINGS

FIG. 1 is a concept view showing a conventional inversion for imaginggeologic structure changing in time.

FIG. 2 is a flowchart illustrating a 4-D inversion and an imaging methodusing it according to one embodiment of the present invention.

FIG. 3 is a concept view showing a 4-D inversion according to oneembodiment of the present invention.

FIG. 4 shows snap images of a changing subsurface space, set for the 4-Dinversion experiment, taken at different times according to oneembodiment of the present invention.

FIGS. 5 and 6 show snap images of a changing subsurface image, taken atdifferent times based on the experiment results using the 4-D inversionaccording to one embodiment of the present invention.

BEST MODE

Hereinafter, 4-D inversion of geophysical data and a 4-D imaging methodof geologic structure using it according to one embodiment of thepresent invention will be explained in detail with reference to theaccompanying drawings.

The present invention is not limited to the following embodiments, butmay be implemented in various ways. The embodiments proposed herein arerather for illustrating the spirit of the invention more sufficientlyfor better understanding.

FIG. 2 is a flowchart illustrating 4-D inversion and an imaging methodusing it according to one embodiment of the present invention.

The present invention relates to the inversion for calculatingdistribution of subsurface material properties from geophysical data andan imaging method using it, which includes:

(a) defining measured data into space-time coordinates, and defining areference space-time model vector (P) and a reference space-time modelvector (U) for a plurality of pre-selected reference times so as todetermine a space-time model (S1);(b) approximating a numerical modeling for a geologic structure spacemodel at an arbitrary time using Taylor series of a numerical modelingfor the plurality of reference space models, defining an objectiveinversion function to give constraints in both space and time domains,and obtaining a reference space-time model vector (U) from the measureddata defined in space-time coordinates using the objective inversionfunction;(c) obtaining a space-time model vector (P) from the referencespace-time model vector (U) to calculate distribution of subsurfacematerial properties in the space-time domain; and(d) imaging geologic structure changing in time, based on thedistribution of subsurface material properties.

The present invention provides a new 4-D inversion capable of invertinga plurality of monitoring data at the same time, and also providingreliable images using not only a plurality of iterative measured databut also only one measured data though subsurface structure is rapidlychanged even during the measurement.

For this purpose, the present invention assumes the subsurface structureas a model that is continuously changed in time, and thus the subsurfacestructure is defined in a space-time model, not a space model. Measureddata are also defined as space-time coordinates, not space coordinates.

If a subsurface space-time model is sampled into a plurality spacemodels at regular time intervals, the subsurface structure will becomposed of numerous space models, so it is substantially impossible toinvert the entire models. As a practical approach to this problem,subsurface space models for pre-selected several reference times(hereinafter, referred to as ‘reference space model’) are set, and it isassumed that material properties at the same space coordinates arelinearly changed in time.

In addition, since numerical modeling for numerous space models alsorequire a lot of calculation, the numerical modeling with respect to asubsurface structure at an arbitrary time is approximated using thefirst order Tayler series of the numerical modeling based on thereference space model.

Using the above assumption and approximation, the question of obtaininga space-time model of geologic structure continuously changing in timecomes to calculating several reference models.

Hereinafter, variables and numerical modeling of geologic structurerelated to the 4-D inversion according to the present invention will beexplained.

If geologic structure continuously changing in time is sampled atregular intervals, the geologic structure will be defined as thefollowing space-time vector (P).

P={p₁, . . . , p_(i), . . . , p_(n)}

Here, p_(i) is a space model vector with respect to time i. Since thespace-time vector P is composed of numerous space models p_(i), a newreference space-time model composed of reference space model vectorsu_(k) with respect to the m (m<<n) number of pre-selected times τ_(k) asfollows is defined.

U={u₁, . . . , u_(k), . . . , u_(m)}

Assuming that subsurface material properties at the same spacecoordinates are linearly changed, the change vector of property changegenerated between two times τ_(k) and τ_(k+1) may be defined as in thefollowing Equation 1 (Math FIG. 1).

$\begin{matrix}{\upsilon_{k} = {\frac{u}{t} = \frac{u_{k + 1} - u_{k}}{\tau_{k + 1} - \tau_{k}}}} & \left( {{Math}\mspace{14mu} {Figure}\mspace{14mu} 1} \right)\end{matrix}$

Using the Equation 1, a subsurface space vector at an arbitrary time tmay be defined as in the Equation 2 (Math FIG. 2).

p(t)=u _(k)+(t−τ _(k))u _(k)  [Math Figure 2]

Though the number of space-domain subsurface space models is reducedusing the above assumption, principally, numerical modeling forcalculating the response of geologic structure according to time shouldbe calculated based on a plurality of subsurface space models. However,it needs a very long calculation time.

In order to solve this problem, the numerical modeling for geologicstructure at an arbitrary time τ_(k)≦t≦τ_(k+1) is approximated using thefirst order Taylor series expansion of F(u_(k)), F(u_(k+1)) that areresults of numerical modeling at both reference times, and then it isused for inversion.

$\begin{matrix}{{G(t)} = {{\frac{\tau_{k + 1} - t}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k} \right)} + {\left( {t - \tau_{k}} \right)J_{k}\upsilon_{k}}} \right\}} + {\frac{t - \tau_{k}}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k + 1} \right)} + {\left( {t - \tau_{k + 1}} \right)J_{k + 1}\upsilon_{k}}} \right\}}}} & \left( {{Math}\mspace{14mu} {Figure}\mspace{14mu} 3} \right)\end{matrix}$

Here, J_(k) is partial derivatives of the model response with respect tothe reference space model u_(k) namely Jacobian matrix.

Adopting the above assumption and approximation, the inversion forfinding 4-D geologic structure over the entire space and time will besimplified into a question for obtaining several reference space modelvectors (u).

Hereinafter, a least-squares inversion related to the 4-D inversionaccording to the present invention will be explained in detail.

An error vector e that is a difference between the measured data vectord and simulation data obtained by the reference space-time model, namelyits data vector, is defined as in the Equation 4 (Math FIG. 4). Inaddition, an estimated error vector e′ that is a difference between atheory data vector obtained by the reference space-time model to beoptimized and calculated in the future and the measured data vector isdefined as in the Equation 5 (Math FIG. 5). The inversion comes to aquestion of calculating an increment vector ΔU of the referencespace-time model so as to minimize the estimated error vector.

e=d−G(U)  [Math Figure 4]

e′=d−G(U+ΔU)  [Math Figure 5]

The inversion of geophysics can be characterized in non-uniqueness andill-posedness of solution, and thus it has serious instability such asdivergence. To solve this problem, constraints are commonly applied tothe inversion. In the present invention, a constraint of time domain isadditionally introduced in addition to the constraint of space domainthat is an inversion constraint commonly adopted to develop an inversionalgorithm of geologic structure in space-time domain. The constraint oftime domain is allowed since definition of the subsurface model isexpanded to space-time domain in the present invention. The 4-Dinversion of the present invention is based on the least-squaresinversion that minimizes squares of error and gives constraints inspace-time domain, so it is defined as a question of minimizing anobjective inversion function Φ expressed in the Equation 6 (Math FIG.6).

Φ=∥e′∥ ²+λΨ+αΓ  [Math Figure 6]

Here, Ψ is an inversion constraint function in space domain, Γ is aninversion constraint function in time domain, and λ and α are Lagrangianmultipliers for controlling the degree of constraints.

A smooth constraint is introduced as the constraint of space domain, andan assumption that there is no serious change between two referencespace models u_(k) and u_(k+1) adjacent in aspect of time is introducedas the constraint of time domain. Such constraints are expressed as theEquations 7 and 8 (Math FIGS. 7 and 8) as follows.

Ψ=(∂^(n) ΔU)^(T)(∂^(n) ΔU)  [Math Figure 7]

$\begin{matrix}\begin{matrix}{\Gamma = {\sum\limits_{i = 1}^{m - 1}{{\left( {u_{k} + {\Delta \; u_{k}}} \right) - \left( {u_{k + 1} + {\Delta \; u_{k + 1}}} \right)}}^{2}}} \\{= {\left\{ {M\left( {U + {\Delta \; U}} \right)} \right\}^{T}{M\left( {U + {\Delta \; U}} \right)}}}\end{matrix} & \left( {{Math}\mspace{14mu} {Figure}\mspace{14mu} 8} \right)\end{matrix}$

Here, M is a square matrix where diagonal elements and one sub-diagonalelement respectively have value 1 or −1.

If the objective inversion function of the Equation 6 is differentiatedwith respect to a reference space-time increment vector ΔU, a solutionfor the reference space-time increment vector ΔU as in the Equation 9(Math FIG. 9) may be obtained.

ΔU=(A ^(T) W _(d) A+C ^(T) ΛC+αM ^(T) M)⁻¹(A ^(T) W _(d) e−αM ^(T)MU)  [Math Figure 9]

Here, W_(d) is a data weighting matrix, A is a partial derivative matrixwith respect to the reference space-time model U composed of referencespace model vectors, and C is a smoothing constraint operator in spacedomain. In case of the constraint of space domain, ACB (ActiveConstraint Balancing) proposed by Yi et al., 2003, is introduced, soLagrangian multipliers for controlling the constraint of space domainare defined as a diagonal matrix A. By solving the Equation 9iteratively until the error between the theory data and the measureddata reaches an allowable limit, a desired geologic structure space-timemodel can be obtained.

FIGS. 1 and 3 are concept views showing the conventional inversion and anew inversion of the present invention, respectively.

In the conventional inversion, monitoring data are separately invertedto realize subsurface images, so it is impossible to refer to changedgeologic structures at different times. Thus, there is high possibilityof distortion of the subsurface images, and the image of subsurfacechange is particularly distorted more seriously. In addition, since eachof monitoring data and subsurface space is analyzed only in a spaceaspect, it is impossible to consider the change of geologic structurethat may happen during the measurement of monitored data. On thecontrary, the 4-D inversion of the present invention analyzes measureddata and geologic structure in both space and time concept, defined intoone kind of space-time domain data and one space-time model, so aspace-time subsurface model may be obtained using one inversion. Due tothis reason, geologic structure changing at different times may bereferred to during the inversion process. In addition, it is possible tocalculate geologic structure changing in time by using just one dataset.

An optimal embodiment has been illustrated in the drawings andspecification. Though specific terms have been used therein, they arefor illustration purpose only and not intended to limit the scope of theinvention defined in the appended claims. Therefore, those havingordinary skill in the art would understand that various modificationsand equivalents may be made therefrom. So, the scope of the presentinvention should be defined based on the spirit of the appended claims.

MODE FOR INVENTION

Hereinafter, inversion experiments to verify effectiveness of the 4-Dinversion and the imaging method using it according to the presentinvention will be explained.

The inversion experiments were conducted using non-conductive tomographynumerical modeling data, pole-dipole array being adopted as an electrodearray, the space-time model being sampled into 30 space models (n=30)continuously changing. Numerical modeling was conducted for all of 30models, and one data set is configured from the data of 30 sets. It isapparent that experiments will be much more difficult in case only onedata set exists rather than in case a plurality of iterative measuredmonitoring data exist, when geologic structure changes severely asabove. This experiment was conducted under the assumption that only onedata set is obtained.

One data set is composed of 1) crosshole, 2) inline, and 3)hole-to-surface surveys.

FIG. 4 shows snapshots of the model change of subsurface space set forthe inversion experiments, taken at each time. The time t is normalizedby the total measurement period.

In the present invention, all data are defined in space-timecoordinates, and numerous space models are obtained from only onemeasurement data set, so the sequence of data measurement will be veryimportant.

FIG. 5 shows snapshots of the model change of subsurface space at eachtime according to the first inversion experiment results.

The inversion experiment shown in FIG. 5 is the inverted results, whendata measurement was performed randomly, and thus the data obtained byeach measurement contain the responses to all the 30 space models. Thenumber of reference models in space domain set to the inversion is onlytwo, and the corresponding pre-selected times are respectively τ=0.17and τ=0.83. Though there are only two reference space models and onlyone data set exists, the inverted results well show the figure that azone with high electric conductivity migrates together with the factthat geologic structure changes in time.

FIG. 6 are snapshots of the model change of subsurface space at eachtime according to the second inversion experiment results.

Differently from FIG. 5 in which measurement was performed randomly,this inversion experiment was conducted based on data configuredsupposing that measurement is done in the sequence of 1) crosshole, 2)inline, and 3) hole-to-surface surveys.

Since the crosshole survey is conducted at an early stage, the responsesof the early time model (see (a) and (b) of FIG. 4) are dominant in thecrosshole survey data. Meanwhile, since the hole-to-surface survey isconducted at a late stage, the responses of the late time model aredominant in the late time model (see (e) and (f) of FIG. 4).

Thus, since geologic structures are different depending on the surveys,it is easily expected that this inversion is significantly difficult incomparison to the case of FIG. 5. FIG. 6 does not show the clearsnapshot images in comparison to FIG. 5, but FIG. 6 still showsreasonable images by which it may be recognized that the geologicstructure is changed during the data collection and the conductive zonemigrates.

In order to examine the influence caused when the number of referencespace models changes, the number of reference space models was set to 3,and then the experiment data used in the calculation of FIG. 6 wasinverted again. However, the results were substantially identical tothose of FIG. 6. Thus, it would be understood that reference spacemodels taken at just two or three times are sufficient in order to imagethe change model of geologic structure continuously changing in time.

In this case, in case of extracting subsurface images using theconventional inversion, only one data set exists, which allows to obtainonly one still subsurface image Thus, it is impossible to extract thechange of geologic structure and also images are seriously distorted.

As described above, the present invention provides a new 4-D inversionof geophysical data and an imaging method using it, which enables toinvert a plurality of monitoring data at the same time and also toconfigure reliable subsurface images even when the change of geologicstructure during data collection cannot be ignored.

By using the inversion of the present invention based on just one-timemeasurement data by means of the crosshole non-conductive tomographysurvey numerical experiments, it is proved that a subsurface imagecontinuously changing in time can be obtained.

Though the embodiments of the present invention have been explained onthe non-conductive surveys, the 4-D inversion of the present inventionmay also be applied to other kinds of physical data such as electronsurveys, gravity surveys, radar, elastic wave tomography and so on,since the 4-D inversion of the present invention has only generalizedconcepts such as numerical modeling and data measurement.

INDUSTRIAL APPLICABILITY

The present invention explained above in detail may be used for a new4-D inversion of geophysical data and a 4-D imaging method of geologicstructure using it. In more detail, the present invention may be usedfor a new 4-D inversion and an imaging method using it, which allows toaccurately calculate geologic structure changing in time by inverting aplurality of monitoring data at the same time, and to provide reliableimages even the geologic structure changes fast during data collection,using only one measurement data, not a plurality of iterativemeasurement data.

1-7. (canceled)
 8. The method for 4-D inversion of geophysical data for calculating distribution of subsurface material properties from geophysical data, comprising: (a) defining measured data in space-time coordinates, and defining a reference space-time model vector (U) composed of a plurality of reference space model vectors (U) for a plurality of pre-selected reference times so as to simulate a space-time model vector (P) that is a geologic structure continuously changing in time; (b) approximating a numerical modeling for a geologic structure space model at an arbitrary time using Taylor series of a numerical modeling for the plurality of reference space models, defining an objective inversion function to constrain each inversion in space and time domains, and obtaining a reference space-time model vector (U) from the measured data defined in space-time coordinates using the objective inversion function; and (c) obtaining a space-time model vector (P) from the reference space-time model vector (U) to calculate distribution of subsurface material properties changing in time.
 9. The method for 4-D inversion of geophysical data according to claim 8, wherein the 4-D inversion is conducted based on the assumption that subsurface material properties are linearly changed in time.
 10. The method for 4-D inversion of geophysical data according to claim 8, wherein a space vector (p(t)) at an arbitrary time (t) is defined using the following equation: p(t)=u _(k)+(t−τ _(k))u _(k)  [Equation] where u is a reference space model vector, ${v_{k} = \frac{u}{t}},$ and τ is a pre-selected reference time.
 11. The method for 4-D inversion of geophysical data according to claim 1, wherein, in the step (b), numerical modeling for the geologic structure space model at an arbitrary time (τ_(k)≦t≦τ_(k+1)) is conducted in a way of approximating by using the first order Taylor series expansion of the following equation for the numerical modeling of reference space models at two reference times (τ_(k), τ_(k+1)): $\begin{matrix} {{G(t)} = {{\frac{\tau_{k + 1} - t}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k} \right)} + {\left( {t - \tau_{k}} \right)J_{k}\upsilon_{k}}} \right\}} + {\frac{t - \tau_{k}}{\tau_{k + 1} - \tau_{k}}\left\{ {{F\left( u_{k + 1} \right)} + {\left( {t - \tau_{k + 1}} \right)J_{k + 1}\upsilon_{k}}} \right\}}}} & \lbrack{Equation}\rbrack \end{matrix}$ where F(u_(k)) is numerical modeling for the reference space model (u_(k)), J_(k) is partial derivatives of a model response with respect to the reference space model, and $v_{k} = {\frac{u}{t}.}$
 12. The method for 4-D inversion of geophysical data according to claim 8, wherein, in the step (b), wherein the objective inversion function uses a function expressed by the following equation: Φ=∥e′∥ ²+λΨ+αΓ  [Equation] where ${e^{\prime} = {d - {G\left( {U + {\Delta \; U}} \right)}}},{\Psi = {\left( {{\partial^{n}\Delta}\; U} \right)^{T}\left( {{\partial^{n}\Delta}\; U} \right)}},\begin{matrix} {\Gamma = {\sum\limits_{i = 1}^{m - 1}{{\left( {u_{k} + {\Delta \; u_{k}}} \right) - \left( {u_{k + 1} + {\Delta \; u_{k + 1}}} \right)}}^{2}}} \\ {{= {\left\{ {M\left( {U + {\Delta \; U}} \right)} \right\}^{T}{M\left( {U + {\Delta \; U}} \right)}}},} \end{matrix}$ Ψ is an inversion constraint function in a space domain, Γ is an inversion constraint function in a time domain, λ and α are Lagrangian multipliers for controlling the degree of constraints, d is a measured data vector, G is a numerical modeling result or a numerical simulation result for a given geologic structure, U is a reference space-time model vector composed of a plurality of reference space model vectors (u), and M is a square matrix where diagonal and one sub-diagonal elements have value 1 or −1.
 13. The method for 4-D inversion of geophysical data according to claim 8, wherein the objective inversion function introduces constraints for the space domain and constraints for the time domain to the inversion at the same time, and, in the constraints for the time domain, the objective inversion function uses a constraint that two reference space models (u_(k), u_(k+1)) adjacent to each other in aspect of time are not greatly changed.
 14. The method for 4-D imaging of geologic structure changing in time, further comprising: imaging geologic structure changing in time based on the distribution of subsurface material properties obtained by the 4-D inversion defined in claim
 8. 15-19. (canceled) 